We also numerically checked the existence of the exactly flat bands in the presence of the \(\mathcal {A}\)  field induced by the lattice relaxation of Ref. [1]}. As demonstrated in Fig. REF , the bandwidth \(W_{band}\)  vanishes when the inter-layer coupling constant \(w_1\)  is tuned to be around \(0.7857 v_F k_{\theta }\) , very close to the value obtained from the approximate formula in Eq. REF . For the BM model where the pseudo-vector field \(\mathcal {A}\)  is absent, the exactly flat bands occur when \(w_1/(v_F k_{\theta }) = 0.586\)  (see Ref.[2]}); if \(w_1\)  and \(v_F\)  are set to the values listed in Table REF  and REF  for the model in Ref. [3]}, the corresponding twist angle is \(1.07^{\circ }\) . However, this angle decreases to \(0.83^{\circ }\)  when the pseudo-vector field \(\mathcal {A}\)  induced by the lattice relaxation [3]} is included in the chiral limit.
